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Register a new userTBG VI: An Exact Diagonalization Study of Twisted Bilayer Graphene at Non-Zero Integer Fillings
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Using exact diagonalization, we study the projected Hamiltonian with Coulomb interaction in the 8 flat bands of first magic angle twisted bilayer graphene. Employing the U(4) (U(4)$times$U(4)) symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent which allows for study around $ u=pm 3,pm2,pm1$ fillings. In the first chiral limit $w_0/w_1=0$ where $w_0$ ($w_1$) is the $AA$ ($AB$) stacking hopping, we find that the ground-states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge $pm1$ excitations found in [arXiv:2009.14200] are the lowest charge excitations up to system sizes $8times8$ (for restricted Hilbert space) in the chiral-flat limit. We also find that the Flat Metric Condition (FMC) used in [arXiv:2009.11301,2009.11872,2009.12376,2009.13530,2009.14200] for obtaining a series of exact ground-states and excitations holds in a large parameter space. For $ u=-3$, the ground state is the spin and valley polarized Chern insulator with $ u_C=pm1$ at $w_0/w_1lesssim0.9$ (0.3) with (without) FMC. At $ u=-2$, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when $w_0/w_1gtrsim0.5t$ where $tin[0,1]$ is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [arXiv:2009.13530]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [arXiv:2009.13530]. For $ u=-3$ with/without FMC, when $w_0/w_1$ is large, the finite-size gap $Delta$ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at $ u=-3$ suggests a competition among (nematic) metal, momentum $M_M$ ($pi$) stripe and $K_M$-CDW orders at large $w_0/w_1$.
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We study magic angle graphene in the presence of both strain and particle-hole symmetry breaking due to non-local inter-layer tunneling. We perform a self-consistent Hartree-Fock study that incorporates these effects alongside realistic interaction and substrate potentials, and explore a comprehensive set of competing orders including those that break translational symmetry at arbitrary wavevectors. We find that at all non-zero integer fillings very small strains, comparable to those measured in scanning tunneling experiments, stabilize a fundamentally new type of time-reversal symmetric and spatially non-uniform order. This order, which we dub the incommensurate Kekule spiral (IKS) order, spontaneously breaks both the emergent valley-charge conservation and moire translation symmetries, but preserves a modified translation symmetry $hat{T}$ -- which simultaneously shifts the spatial coordinates and rotates the $U(1)$ angle which characterizes the spontaneous inter-valley coherence. We discuss the phenomenological and microscopic properties of this order. We argue that our findings are consistent with all experimental observations reported so far, suggesting a unified explanation of the global phase diagram in terms of the IKS order.
We derive the exact insulator ground states of the projected Hamiltonian of magic-angle twisted bilayer graphene (TBG) flat bands with Coulomb interactions in various limits, and study the perturbations away from these limits. We define the (first) chiral limit where the AA stacking hopping is zero, and a flat limit with exactly flat bands. In the chiral-flat limit, the TBG Hamiltonian has a U(4)$times$U(4) symmetry, and we find that the exact ground states at integer filling $-4le ule 4$ relative to charge neutrality are Chern insulators of Chern numbers $ u_C=4-| u|,2-| u|,cdots,| u|-4$, all of which are degenerate. This confirms recent experiments where Chern insulators are found to be competitive low-energy states of TBG. When the chiral-flat limit is reduced to the nonchiral-flat limit which has a U(4) symmetry, we find $ u=0,pm2$ has exact ground states of Chern number $0$, while $ u=pm1,pm3$ has perturbative ground states of Chern number $ u_C=pm1$, which are U(4) ferromagnetic. In the chiral-nonflat limit with a different U(4) symmetry, different Chern number states are degenerate up to second order perturbations. In the realistic nonchiral-nonflat case, we find that the perturbative insulator states with Chern number $ u_C=0$ ($0<| u_C|<4-| u|$) at integer fillings $ u$ are fully (partially) intervalley coherent, while the insulator states with Chern number $| u_C|=4-| u|$ are valley polarized. However, for $0<| u_C|le4-| u|$, the fully intervalley coherent states are highly competitive (0.005meV/electron higher). At nonzero magnetic field $|B|>0$, a first-order phase transition for $ u=pm1,pm2$ from Chern number $ u_C=text{sgn}( u B)(2-| u|)$ to $ u_C=text{sgn}( u B)(4-| u|)$ is expected, which agrees with recent experimental observations. Lastly, the TBG Hamiltonian reduces into an extended Hubbard model in the stabilizer code limit.
The dominance of Coulomb interactions over kinetic energy of electrons in narrow, non-trivial moir{e} bands of magic-angle twisted bilayer graphene (TBG) gives rise to a variety of correlated phases such as correlated insulators, superconductivity, orbital ferromagnetism, Chern insulators and nematicity. Most of these phases occur at or near an integer number of carriers per moir{e} unit cell. Experimental demonstration of ordered states at fractional moir{e} band-fillings at zero applied magnetic field $B$, is a challenging pursuit. In this letter, we report the observation of states at half-integer band-fillings of $ u = 0.5$ and $3.5$ at $Bapprox 0$ in a TBG proximitized by a layer of tungsten diselenide (WSe$_2$). The magnetotransport data enables us to deduce features in the underlying band structure consistent with a spontaneously broken translational symmetry supercell with twice the area of the original TBG moir{e} cell. A series of Lifshitz transitions due to the changes in the topology of the Fermi surface implies the evolution of van Hove singularities (VHS) of the diverging density of states at a discrete set of partial fillings of flat bands. Further, we observe reset of charge carriers at $ u = 2, 3$. In addition to magnetotransport, we employ thermoelectricity as a tool to probe the system at $B=0$. Band structure calculations for a TBG moir{e} pattern, together with a commensurate density wave potential and spin-orbit coupling (SOC) terms, allow to obtain degeneracy-lifted, zone-folded moir{e} bands with spin-valley isospin ordering anisotropy that describe the states at half-integer fillings observed experimentally. Our results suggest the emergence of a spin-charge density wave ground state in TBG in the zero $B-$ field limit.
The tunable magnetism at graphene edges with lengths of up to 48 unit cells is analyzed by an exact diagonalization technique. For this we use a generalized interacting one-dimensional model which can be tuned continuously from a limit describing graphene zigzag edge states with a ferromagnetic phase, to a limit equivalent to a Hubbard chain, which does not allow ferromagnetism. This analysis sheds light onto the question why the edge states have a ferromagnetic ground state, while a usual one-dimensional metal does not. Essentially we find that there are two important features of edge states: (a) umklapp processes are completely forbidden for edge states; this allows a spin-polarized ground state. (b) the strong momentum dependence of the effective interaction vertex for edge states gives rise to a regime of partial spin-polarization and a second order phase transition between a standard paramagnetic Luttinger liquid and ferromagnetic Luttinger liquid.
Motivated by the recently observed insulating states in twisted bilayer graphene, we study the nature of the correlated insulating phases of the twisted bilayer graphene at commensurate filling fractions. We use the continuum model and project the Coulomb interaction onto the flat bands to study the ground states by using a Hartree-Fock approximation. In the absence of the hexagonal boron nitride substrate, the ground states are the intervalley coherence states at charge neutrality (filling $ u$ = 0, or four electrons per moire cell) and at $ u$ = -1/4 and -1/2 (three and two electrons per cell, respectively) and the $C_2mathcal{T}$ symmetry-broken state at $ u$= -3/4 (one electron per cell). The hexagonal boron nitride substrate drives the ground states at all $ u$ into $C_2mathcal{T}$ symmetry broken-states. Our results provide good reference points for further study of the rich correlated physics in the twisted bilayer graphene.
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